UPDATE: This answer may now be outdated, see comments below. This was going to happen eventually anyway, as YouTube phases out FLV and shifts toward HTML5 video...
They simply resolve the link to the actual FLV file, and download it.
This is done by copying the video identifier from the URL:
http://www.youtube.com/watch?v=WEeqHj3Nj2c
Which is used to request info about the video:
http://www.youtube.com/get_video_info?&video_id=WEeqHj3Nj2c
The video info includes a TOKEN, which you can then you use to make another request:
http://www.youtube.com/get_video?video_id=WEeqHj3Nj2c&t=TOKEN
If all goes well, YouTube will respond with HTTP 303 See Other, including a Location header with a direct link to the FLV file.
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